N O T I C E THIS DOCUMENT HAS BEEN REPRODUCED FROM MICROFICHE. ALTHOUGH IT IS RECOGNIZED THAT CERTAIN PORTIONS ARE ILLEGIBLE, IT IS BEING RELEASED IN THE INTEREST OF MAKING AVAILABLE AS MUCH INFORMATION AS POSSIBLE /) NASA TECHNICAL MEMORANDUM NASA TM 75732 SOLUTION OF A FEW NONLINEAR PROBLEMS IN AERODYNAMICS BY THE FINITE ELEMENTS AND FUNNCTIONAL LEAST SQUARES METHODS Jacques Periaux Translation of "Resolution de quelques problemes non lineaires en aerodynamique par des methodes d'elements finis et le moindres carres € fonctionnels, It Doctoral Dissertation, University of Paris, June 19, 19%9, 280 pages (NASA-TM-75732) SOLUTION OF A FEW NONLINEAR N80-17990 PROBLEMS IN AERODYNAMICS BY THE FINIT ELEMENTS AND FUNCTIONAL LEAST SQUARES METHODS Ph.D. Thesis - Paris (National Unclas Aeronautics and Space Adainistration) 278 p G3/02 47187 ^ "0 I NATIONAL AI.AZO.NAUTTCS AND SPAC!; ELI MINI ''."_1ATION WASHT\GTO\, D.C. 20546 DECEMBER 1979 AIL- ;; ACKNOWLEDGMENTS a This thesis is the result of two years of applied research un- der the direction of Professor R. GLOWINSKI and is the fruit of a research collaboration between the LABORIA/IRIA and the AMD/BA in- dustries. I should like to thank Professor J.L. LIONS whose instruction made it possible to undertake this research. I am deeply indebted to Professor R. GLOWINSKI who directed and encouraged this research, and who gave me the honor of presiding over the examining committee. I should like to acknowledge my deepest gratitude to MR. 0. PIRONNEAU, who together with MR. GLOWIN 1"KIq originated the optimal control methods presented, and who lavished me with advice and en- couragement throughout this research. I thank Professor P.A. r.AVIART for giving me the honor of being a member of the examining committee. Special thanks to Mr. P. BOHN t director of the Advanced Studies Division (Division des Etudes Avancees), and to Mr. P. PERRIERv Chief of the Department of Aerodynamic Theory (Departement d'Aerody- namique Theorique)p and to the AAiD/BA for participating on the exam- ining commitee and for their knowledge of numerical methods and of powerful computer centers which was shared with me. I should like to acknowledge other debts: the discussions with Mr. PERRIER and his helpful suggestions in tiie field of Fluid Mech- anics, and the discussions with Mrs. M.O. BRISTEAli du LAPORIA and Mr Mr. Be MANTEL and Mr. G. POIRIER from AMD/BA. I should also like to thank them for their friendly collaboration throughout this study. My thanks go also to Mrs. F. WEBER and Mr. HUBERT for the care- ful typing and drawings of this document. Finally, this thesis is dedicated to my wife and to the memory of her father. ^,. Ntimbera in the margin indicate pagination in the foreign text. '-"'c CEMVG P AGE 81 AIIK Nor F'LME. SUMMARY L The objective of this study is to provide the numerical sim- ulation of the transsonic flows of idealized fluids and of incom- pressible viscous fluids, by the non linear least squares methods of R. GLOWINSKI and 0. PIRONNEAU. The complexity o: the geometries studied in industrial aerodynamics explains the preference given to the finite elements for the approximation of the equations. Chapters 1, 2, 3 9 4 describe the non linear equations, the boundary conditions and the various constraints controlling the two types of flow. The standard iterative methods for solving a quasi elliptical non linear equation with partial derivatives (E.D. P.) are briefly reviewed in Chapter 5 with emphasis placed on two examples ; the fixed point method applied to the Gelder functional in the case of compressible subsonic flows and the Newton method used in the technique of decomposition of the lifting potential. Chapter 6 presents the new abstract least squares method. It consists of substituting the non linear equation by a problem of minimization in a H- 1 type Sobolev functional space, which is itself equivalent to an optimal control problem and solved by a conjugate gradient algorithm with metric H l . The application of this method- ology to transsonic equations is presented in Chapter 7. We show how to include within the optimal control formulation two con- straints of aerodynamics: the condition of entropy, on the one hand, treated either by penalization or by artificial viscosity, and the Joukowski condition, on the other hand, taken into account by a fix- ed point method on circulation. The Navi.er-Stokes equations are reduced to a problem of minimi- zation in 11- 1 in the same manner in Chap ter 8. Accordingly, we show that the state systems of the mixed optimal control problem are generalized Stokes problems in steady and unsteady cases, after quantification in time with the use of implicit Crank-Nicholson (for example) type schemes. To solve them, a mixed formulation proposed '_^y GLOWINSKI-PIRONNEAU and 'cased on certain decomposition properties of the biharmonic operator, is used. The Stokes algo- L"' rithm is substitued by a sequence of Dirichlet problems coupled with an integral equationE conditioned on the pressure trace, defined on the boundary of the domain occupied by the fluid. Chapters 9 and 10 are devoted to the approximation of a. trans- sonic and Navier-Stokes optimal control formulation by PIc Lagrange conform Finite elements, with de,-roe k=1 or 2. The numeric<<1 imple- mentation of the conjugate Gradient algorithms is developed and presented in the form of flow charts. The numerical implementa- tion cf the Stokes algorithm (LI ) is described and the choice of a uirect (Cho.loshi) or iterative ^prccondi.tioned conjugate jradient moUjod for solvi iii; it is discus-sod. `i`he large at:nounts of computations, clue to complox trill i;nci;- sional confi;-urations (:1.1celle t vehicle, air-inlet, ..irplr^nc^^, storey in the main core of the computor, require an incomplete Choleski factorization of the discrete Dirichlet matrices shown on the inside of the control. loop, The use of auxAliary operators LLt in the solution of an optimal control problem is presented in Chapter 11 through comparisons of rosearch results of J.A. MEIJ- ERFINK-M.A. VAN DER VORST and 0. AXELSS0N. The numerical experiments are described in Chapter 12. The transsonic calculations obtained from the finite elements-optimal control codes are compared with those obtained from the finite differences codes of A. JAMESON on a NACA 0012 airfoil and a Kern airfoil. More complex transsonic configurations of industrial aerody- namics such as multi-bodies or air inlets u,e analyzed. The feasibility of optimal control conjugate gradient algo- rithms is verified on bi and tridimensional Navier-Stokes calcu- lations requiring considerable data processing resources (memory and CPUJ. Separated flows around/in an air inlet and around an swept-back wing with high incidence, are simulated numerically by following at various time cycles the evolution of the field of velocities, the field of pressures, the streamlines and the vorti- city. Finally, the last paragraph of Chapter 1 2 is devoted to the data processing efficiency of the auxiliary, operators. It shows, through examples taken from the two flow families, how it is pos- sible, by using preconditioned optimal control algorithms, to cal- culate entirely in the main core of the computer, with small per- centages of Dirichlet matrices A d/170 (5`d<20) without reducing the convergence velocity of the algorithm. TABLE OF CONTENT'S Pnges 0. INTRODUCTIO N 0.1 Applications of non linear aerodynamics to the aero- nautics industry 0.2 Difficulties rolating to industrial configurations 1 1. FEASIBLE MODELING OF AN INCOMPRESSIBLE IDEALIZED FLUID 2 FLOW 1.1 Non lifting case 2 1.2 Lifting case 5 2. FEASIBLE MODELING OF A SUBSONIC COMPRESSIBLE IDEALIZED FLUID FLOW 8 2.1 Non lifting case g 2.2 Lifting case U 3. MODI~i_TNG nF ruF. PQ rr%TTTAT Cn1=rr , FjnW rgn*kLe rnM= FAESSIBLE IDEALIZED FLUID 9 3.1 Equations 9 3.2 The condition of entropy formulated as a constraint 11 3.3 The condition of entropy formulated by artificial viscosity 13 3.4 Lifting case 14 4. MODELING VELOCITY-PRESSURE OF AN UNSTEADY INCOMPRESSIBLE VISCUOUS FLUID FLOW Navier-Stokes 14 5. STANDARD ITERATIVE METHODS FOR SOLVING QUASI ELLIPTICAL NO1\ ELLIPTICAL E„D.P. lE 5.1 The model problem 16 5.2 The fixed point methods (Gelder all;orithm) or qunsi- linearization 16 5.3 Newton methods 20 5.3.1. Lifting incompressible fluid i_ow around a body 20 5.3.2. Expansion of a lifting subsonic compressible fluid flow around a multibody by the technique of decomposition. ^4 5.4 The pseudo-unstead y methods(Arrow-Hurwicr algorithm) ;,^; ti. 1'1311 F'1 1 NCT1- 0%,A1_ I. IIAST SgUARES METHODS 21cl 6.1 Relationships between a toast sgtttres method and att optimal control problvm The least 5quart•s method in .1 p;articular fitnetiollal space: 11 -1 '}U 0.1 Iterative saltation of .111 optimal control problor» 1)N • is con jilgate 1-radj olit all-ori than ;,' tt 7• THE LEAST SQUARES METHOD IN H -1 APPLIED TO TRANSSONIC FLOWS 34 Z 7.1. The subsonic non lifting case 34 7.2. The transsonic non lifting case 34 7.3. The transsonic lifting case 35 7.4. Conjugate gradient solution of the transsonic problem 39 8. THE LEAST SQUARES METHOD IN H -1 APPLIED TO THE NAVIER-STOKES 41 EQUATION= 8.1. The steady case 41 8.1.1. Functional least squares method of steady N.S. 41 equations 80.20 Conjugate gradient solution of steady N.S. equa-42 tions 8.2. The unsteady case 8.2.1. Formulation of the Naviei-Stokes unsteady pro- 45 blem 45 8.2.2.